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Opers versus nonabelian Hodge

Jul 7, 2016
23 pages
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Abstract: (submitter)
For a complex simple simply connected Lie group GG, and a compact Riemann surface CC, we consider two sorts of families of flat GG-connections over CC. Each family is determined by a point u{\mathbf u} of the base of Hitchin's integrable system for (G,C)(G,C). One family ∇ℏ,u\nabla_{\hbar,{\mathbf u}} consists of GG-opers, and depends on ℏ∈C×\hbar \in {\mathbb C}^\times. The other family ∇R,ζ,u\nabla_{R,\zeta,{\mathbf u}} is built from solutions of Hitchin's equations, and depends on ζ∈C×,R∈R+\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+. We show that in the scaling limit R→0R \to 0, ζ=ℏR\zeta = \hbar R, we have ∇R,ζ,u→∇ℏ,u\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}. This establishes and generalizes a conjecture formulated by Gaiotto.